On A Generalisation Of Signed Dominating Functions Of Graphs

For a positive integer \(k\), a \(k\)-subdominating function of \(G = (V, E)\) is a function \(f: V \to \{-1, 1\}\) such that the sum of the function values, taken over closed neighborhoods of vertices, is at least one for at least \(k\) vertices of \(G\). The sum of the function values taken over all vertices is called the aggregate of \(f\) and the minimum aggregate amongst all \(k\)-subdominating functions of \(G\) is the \(k\)-subdomination number \(\gamma_{ks}(G)\). In the special cases where \(k = |V|\) and \(k = \lceil|V|/2\rceil\), \(\gamma_{ks}\) is respectively the signed domination number [{4}] and the majority domination number [{2}]. In this paper we characterize minimal \(k\)-subdominating functions. By determining \(\gamma_{ks}\) for paths, we give a sharp lower bound for \(\gamma_{ks}\) for trees. We also determine an upper bound for \(\gamma_{ks}\) for trees which is sharp for \(k \leq |V|/2 \).